The evolving slope of the stellar mass function at 0.6 <= z < 4.5 from deep WFC3 data

0.73 MB·English

by  P. Santini

Astronomy&Astrophysicsmanuscriptno.santini (cid:13)c ESO2012
January11,2012
The evolving slope of the stellar mass function at 0.6 ≤ z < 4.5
from deep WFC3 data
P.Santini1,A.Fontana1,A.Grazian1,S.Salimbeni2,F.Fontanot3,D.Paris1,K.Boutsia1,M.Castellano1,F.Fiore1,S.
Gallozzi1,E.Giallongo1,A.M.Koekemoer4,N.Menci1,L.Pentericci1,andR.S.Somerville4,5
1 INAF-OsservatorioAstronomicodiRoma,viadiFrascati33,00040MontePorzioCatone,Italy.
2 AstronomyDepartment,UniversityofMassachusetts,Amherst,MA01003,U.S.A.
3 INAF-OsservatorioAstronomicodiTrieste,ViaG.B.Tiepolo11,34131Trieste,Italy.
2 4 SpaceTelescopeScienceInstitute,3700SanMartinDrive,Baltimore,MD21218.
1 5 DepartmentofPhysicsandAstronomy,JohnsHopkinsUniversity,Baltimore,MD21218.
0
Received....;accepted....
2

n ABSTRACT
a
J WeusedEarlyReleaseScience(ERS)observationstakenwiththeWideFieldCamera3(WFC3)intheGOODS-Sfieldtostudythe

0 galaxystellarmassfunction(GSMF)at0.6 ≤ z < 4.5.DeepWFC3near-IRdata(forY asfaint as27.3, J and H asfaintas27.4
1 ABmagat5σ),aswellasdeepKS (asfaintas25.5at5σ)Hawk-Ibanddata,provideanexquisitedatasetwithwhichdeterminein
anunprecedentedwaythelow-massendoftheGSMF,allowinganaccurateprobeofmassesaslowas M∗ ≃ 7.6·109M⊙ atz ∼ 3.
] Althoughtheareausedisrelativelysmall(∼33arcmin2),wefoundgenerallygoodagreementwithpreviousstudiesontheentiremass
O range.Ourresultsshowthattheslopeofthefaint-endincreaseswithredshift,fromα=−1.44±0.03atz∼0.8toα=−1.86±0.16at
C z∼3,althoughindicationsexistthatitdoesnotsteepenfurtherbetweenz∼3andz∼4.Thisresultisinsensitivetoanyuncertainty
in the M∗ parameter. The steepness of the GSMF faint-end solves the well-known disagreement between the stellar mass density
.
h (SMD)andtheintegratedstarformationhistoryatz>2.However,weconfirmthethatthereappearstobeanexcessofintegratedstar
p formationwithrespecttotheSMDatz<2,byafactorof∼2−3.Ourcomparisonoftheobservationswiththeoreticalpredictions
- showsthatthemodelsforecastagreaterabundanceoflowmassgalaxies,atleastuptoz∼3,aswellasadearthofmassivegalaxies
o atz∼4withrespecttothedata,andthatthepredictedSMDisgenerallyoverestimatedatz.2.
r
t Key words. Galaxies: luminosity function, mass function - Galaxies: evolution - Galaxies: high-redshift - Galaxies: fundamental
s
a parameters
[

2
v 1. Introduction ysis to z & 3 (e.g. Fontanaetal. 2006; Kajisawaetal. 2009;
8 Caputietal.2011).Inparallel,wide-fieldsurveyshaveprovided
Understandingtheassemblyofstellarmassingalaxiesisafun-
2 large samples with more accurate statistics (Droryetal. 2009;
damental step towards a description of galaxy evolution. Key
7 Pozzettietal. 2010; Bolzonellaetal. 2010; Marchesinietal.
tools to study this process through cosmic time are the galaxy
5 2010;Ilbertetal.2010).Oneofthekeyresultsofthesesurveys
1. stellar mass function(GSMF) and its integralover masses (the hasbeenthedemonstrationthattheshapeoftheGSMFcannot
stellarmassdensity,SMDhereafter).
1 bedescribedbya(widelyadopted)singleSchechterfunctionat
Therefore,itisnosurprisethatmostextragalacticsurveysin
1 leastuptoz ≃ 1.5,butthatitdepartsfromthisparametricform
thepastdecadehavebeenusedtodeterminetheshapeandevo-
1 because of the superposition of individual distributions for the
lutionoftheGSMFasafunctionofredshift.Theearliestresults
: redandbluegalaxypopulations(Ilbertetal.2010;Pozzettietal.
v based on small field surveys revealed that the SMD decreases
2010; Bolzonellaetal. 2010; Mortlocketal. 2011) or/and be-
i
X withincreasingredshift,asexpectedintheframeworkofthecur- causeofachangewithstellarmasseitherinstarformationeffi-
rentlyacceptedcosmologicalhierarchicalscenario,bothinterms
r ciencyorgalaxyassemblyrate(Droryetal.2009).
a of integrated SMD (e.g. Giallongoetal. 1998; Dickinsonetal.
AnaccurateknowledgeoftheGSMFisalsoasensitivetest
2003; Fontanaetal. 2003; Rudnicketal. 2003), as well as of
of modern galaxy evolutionary models. From its initial stud-
the detailed GSMF (e.g.Fontanaetal. 2004; Droryetal. 2004,
ies, much interest in the GSMF has been triggeredby the pos-
2005). There have since been many indications that the evo-
sibility of constraining the physics of the evolution of more
lution of the GSMF occurs more rapidly for more massive
massive galaxies, which, according to the hierarchical struc-
galaxies than for low mass ones (e.g. Fontanaetal. 2006;
tureformationscenario,arethe resultsofthemergingoflower
Pozzettietal.2007;Pe´rez-Gonza´lezetal. 2008;Kajisawaetal.
mass objects at earlier times (Coleetal. 1994). In addition,
2009; Marchesinietal. 2009), a behaviour known as downsiz-
to achieve a complete view of the galaxy formation picture,
ing in stellar mass (see Fontanotetal. 2009, and references
an important goal is a robust knowledge of the properties of
therein). The adventof near- and mid-infraredfacilities, above
low mass galaxies at high redshift. The slope of the GSMF at
alltheSpitzertelescope,hasallowedtheuncertaintiesinstellar
lowmassesmayalsorepresentacriticalbenchmarkforcurrent
mass estimates to be reduced and the extension of their anal-
galaxy formation models. There is growing evidence that the
Send offprint requests to: P. Santini, e-mail: number of low mass galaxies in the Universe is systematically
[email protected] overpredicted by most or all theoretical models (Fontanaetal.
1
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
2006; Fontanotetal. 2009; Marchesinietal. 2009). Very sim- reconcilethe two observables(Fardaletal. 2007; Wilkinsetal.
ilar evidence also appears in the analysis of the luminosity 2008). However, before invoking the non-universality of the
functions (Polietal. 2001; Marchesini&vanDokkum 2007; IMF, it must be noted that both the SFRD and the SMD are
LoFaroetal.2009;Henriquesetal.2011).Aparticularlystrik- affected by large uncertainties. The measure of the star forma-
ing aspect of the mismatch is that it appears in different ren- tion rate is itself particularly difficult, being either highly de-
ditions of theoretical models, suggesting that it marks some pendent on uncertain dust corrections (e.g. Santinietal. 2009;
fundamentalincompletenessinourtheoreticalunderstandingof Nordonetal.2010)orlimitedtothebrightestfar-infraredgalax-
galaxyformationandgrowth. ies at z < 2−3 (Rodighieroetal. 2010). On the SMD side, in
Whileaglobalpictureisemergingfromtheseinvestigations, additiontothe uncertaintiesrelatedto thestellar massestimate
many outstanding questions are still to be addressed. In gen- itself,amajorroleisplayedbythepoorknowledgeofthelow-
eral,thevariousGSMFspresentedintheliteratureagreereason- mass tail of the GSMF. Owing to the limited depths of current
ablywellatz = 0−5,althoughdisagreementsexist,somewhat IR surveys, the estimate of the faint-end slope basically relies
increasing at high redshift (Caputietal. 2011; Gonza´lezetal. on large extrapolations. An incorrect estimate, given the large
2011;Marchesinietal.2010;Mortlocketal.2011),thatcannot number density of low mass objects, could translate into non-
be explained by merely field-to-field variance. At even higher negligibleerrorsintheSMD.
redshift, the available estimates of the SMD are based on UV- ArobustestimateoftheslopeoftheGSMFisnecessaryto
selected samples, hence are potentially incomplete in mass, providetighterconstraintson allthese unknowns.Inthisstudy
and/orareoftenderivedbyadoptingaveragemass-to-lightratios we take advantageofthe recentdeepnear-IRobservationscar-
for the whole population rather than detailed object-by-object riedoutbyWideFieldCamera3(WFC3)installedontheHSTin
estimates (Gonza´lezetal. 2011). Finally, and particularly rele- theupperpartoftheGOODS-SfieldintheY,JandHbandsand
vantforthemaintopicofthispaper,theGSMFatlowmassesis byHawk-ImountedatVLTintheK band.Thesedataallowac-
S
highlyuncertainatintermediateandhighredshifts,sincecurrent curatemeasurementsofthestellarmasstoverylowlimits.Inthis
samplesdonotextendtothedepthsrequiredtoestablishitsslope respect,weextendtohigherredshiftsandlowermassesthedeep
withgoodaccuracy.Theseuncertaintiesareduetoanumberof analysis carried out by Kajisawaetal. (2009). The only study
observationallimitations. of comparable depth is Mortlocketal. (2011), which was also
InadditiontotheuncertaintiesrelatedtotheGSMFcompu- based on WFC3 data. However, the greater depth of the Early
tation,itmustnotbeforgottenthattheactualestimatesofstellar ReleaseScience(ERS)imagesusedinthisworkandtheconser-
massesfrombroadbandphotometryarepotentiallyaffectedby vativecutsthatweapplytothesampleensureanexcellentover-
manysystematicuncertainties,evenwhenaccurateredshiftsare allphotometricquality,aswediscussinSect.3.3.Unfortunately,
available. Part of this uncertainty is due to the lack of knowl- the areacoveredbyERS observationsissmallcomparedtore-
edge of importantparametersof the stellar population,such as cent surveys, and is slightly overdense.This feature somewhat
metallicity or extinctioncurve.The modellingof highlyuncer- limits the universal validity of our results regarding the SMD,
tainphasesofstellarevolutionisanothersourceofuncertainty: especially in the intermediate redshift bins, although we chose
in particular the differenttreatments of the thermally pulsating our redshift intervals in order to ensure that the known clus-
asymptotic giant branch (TP-AGB) phase is the source of the ters and groups (discussed in Sect. 2) were mostly confined to
highest discrepancies in simple stellar population models (see two of them. However,we show thatthe study ofthe faint-end
e.g.Maraston2005;Marigoetal.2008),andhasrelevantimpli- slope,whichisthemainaimofthepresentanalysis,isinsignif-
cations for the estimate of near-infrared luminosities and stel- icantly affected by these cosmic variance effects. In addition,
lar masses for galaxies dominated by intermediate-age stellar this work represents an exercise to explore the potential of fu-
populations (∼ 1 Gyr). The largest bias is due to the difficul- turedeepWFC3 observations,suchas thoseof the CANDELS
ties in reconstructingthe star formationhistoryof each galaxy, survey(Groginetal. 2011; Koekemoeretal. 2011), which will
which is necessary to estimate the appropriate M /L ratio, and coveramuchmoreextendedareaovervariousfieldswithdepths
∗
thatmaybepoorlydescribedbysimplisticmodelssuchasthose comparabletotheERSobservations.
adopted in stellar population synthesis codes (Marastonetal. Thepaperisorganizedasfollows:afterintroducingthedata
2010;Leeetal.2010). inSect.2,wepresentthestellarmassestimateandtheGSMFin
All these uncertainties contribute to one of the main puz- Sect. 3, the analysis of the faint-endslope in Sect. 4, the SMD
zles that appear in present-day observational cosmology: the anditscomparisonwiththeintegratedSFRDinSect.5,andthe
mismatch between the observed SMD and the integrated star comparisonwiththeoreticalpredictionsinSect.6.Sect.7sum-
formation rate density (SFRD) (e.g. Hopkins&Beacom 2006; marizesourresults.Inthefollowing,weadopttheΛ-CDMcon-
Fardaletal. 2007; Wilkinsetal. 2008). In principle, these two cordance cosmological model (H0 = 70 km/s/Mpc, ΩM = 0.3
observables represent independent approaches to studying the andΩΛ=0.7).AllmagnitudesareintheABsystem.
mass assembly history from differentpoints of view. However,
the integrated star formation, after considering the gas recycle
2. Thedatasample
fractioninto the interstellar medium,appearsto be higherthan
the observed SMD at all redshifts. Several authors have high- Thisworkexploitsanewsetofnear-IRimagesthatrepresenta
lightedthisseverediscrepancy(ofuptoafactorof∼4atz∼3, significant improvementin photometricquality and depth over
Wilkinsetal. 2008). Moreover, if the merging contribution to existingsurveys.Thefirstcomponentisthepublicreleaseofthe
the stellar mass build-up is accounted for (Drory&Alvarez Early Release Science (ERS) observations taken with WFC3,
2008), the agreement gets even worse. Intriguingly, the inte- the new near-IR camera on board HST. The ERS observations
grated SFRD exceedsthe observedSMD, implying that we ei- coveranareaof∼50arcmin2,locatedinthenorthern∼30%of
ther overestimate the SFRD, or miss a substantial fraction of the GOODS-South field. They were taken in three filters, Y ,
098
massivegalaxies,orunderestimatetheirmasses,orfinallyfailin J , and H , which reach 27.3 (Y) and 27.4 (J, H) magni-
125 160
reconstructing the low-mass tail of the GSMF. An initial mass tudesat5σinanareaof∼0.11arcsec2.We usedtheERSmo-
function (IMF) that varies over cosmic time was invoked to saicsproducedasdescribedinKoekemoeretal.(2011);wealso
2
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
refer to Grazianetal. (2011) for details of the catalogues, and
Windhorstetal.(2011) fora fulldescriptionof theERS obser-
vationalprogram.
WecomplementedtheseimageswithnewdeepK bandim-
S
agestakenovertheGOODS-Sfieldwiththenear-IRVLTimager
Hawk-I.Thelatterweretakenintheframeworkofaprogramde-
signedtosearchforz∼7galaxies(Castellanoetal.2010a,b).In
the K band, the surveyedarea covers80%of the WFC3 ERS
S
area.Owingtothisandafterexcludingimageedgesofdubious
quality,theavailableareareducesto∼33arcmin2.Thedatare-
ductionoftheK imagesisanalogoustotheprocedureusedfor
S
other Hawk-I data (Castellanoetal. 2010a). The net exposure
time is25200s,witha 1σr.m.s. of1.26countspersecondina
1”aperture.Themagnitudelimitat5σis∼25.5,onemagnitude
deeperthanthepreviousISAACK band.
S
We finally builta multiwavelengthGOODS-ERScatalogue
adding the other public images available in the GOODS-
S field. They include the ACS images in the BVIz bands Fig.1. Redshift distribution of the K ≤ 25.5 GOODS-ERS
S
(Giavaliscoetal. 2004), the deep UR images from VIMOS sample (black solid histogram) compared to the K ≤ 23.5
S
(Noninoetal. 2009) and the four IRAC bands at 3.6, 4.5, GOODS-MUSIConeadoptedbyFontanaetal.(2006)(reddot-
5.8, and 8.0 µm. With respect to the data set used to assem- ted histogram). Overdensities at z ≃ 0.7, z ≃ 1, z ≃ 1.6, and
ble ourpreviousGOODS-MUSICsample (Grazianetal. 2006; z ≃ 2.2−2.3canberecognizedinthedistribution(seetextfor
Santinietal. 2009), the present GOODS-ERS data set benefits references).
notonlyfromthemuchdeeperIRcoverageprovidedbythenew
WFC3 and K banddata,butalsofromadeeperversionofthe
S
z band image, which nearly doubles the exposure time of the
previousimage,adeeperU bandimage,andabrandnewRim- theK bandarealsodetectedintheHone,whichisunsurprising
S
age. In this data set, we extracted a 14 band multiwavelength giventheextraordinaryqualityoftheWFC3data.
catalogueusingthe H bandasadetectionimage.Colourswere Onthebasisoftheseresults,wedecidedtorestrictouranaly-
carefullyobtainedwiththesametechniqueusedintheGOODS- sistotheK ≤25.5sample,albeitobtainedfromtheH-selected
S
MUSIC catalogue, where we adopted the PSF-matching code one,fortworeasons:firstly,thisselectionallowsamorerobust
CONVPHOT (DeSantisetal. 2007) to accurately deblend ob- comparison with previous K-selected surveys; and secondly, a
jects in the ground-basedand Spitzer images. We note that the K = 25.5 threshold is more efficient in detecting low mass
S
depthoftheH bandevenexceedsthedepthofthebluestbands, objects than a H = 26 one. Adopting this cut, we extend by
resultinginverypoorqualityphotometricinformationaboutthe twomagnitudesthepreviousworkofFontanaetal.(2006),who
faintestH-selectedobjects. studied the GSMF of the K < 23.5 GOODS-MUSIC sample.
S
OurK ≤25.5samplehereincludes3210objects,421ofwhich
The catalogue was cross-correlated with existing spectro- S
havespectroscopicredshifts.
scopic samples. For sources lacking spectroscopic informa-
tion, photometric redshifts were computed by fitting the 14 We plot in Fig. 1 the redshift distribution of the GOODS-
bandmultiwavelengthphotometrytothePEGASE2.0templates ERSsampleusedinthiswork(blacksolidhistogram)compared
(Fioc&Rocca-Volmerange 1997, see details in Grazianetal. tothatoftheGOODS-MUSICsampleadoptedbyFontanaetal.
2006).Theaccuracyreachedbythephotometricredshiftsisvery (2006)(reddottedhistogram).SincetheareacoveredbytheERS
high,theabsolutescatter|∆z|/(1+z )beingequalto0.03,with surveyis relativelysmall, the sampleis moresensitiveto over-
spec
only3%ofsevereoutliers(|∆z|/(1+z ) > 0.5).Thestatisti- densities. The extended overdensities at z ≃ 0.7 and z ≃ 1,
spec
calerrorassociatedwith eachphotometricredshiftwas usedto which cover the entire GOODS-S field (Vanzellaetal. 2005,
evaluate the limiting magnitude at which a reliable GSMF can Salimbenietal. 2009a and references therein), are clearly rec-
be computed. We found that a limit H ≃ 26 (or, equivalently, ognizable. Unfortunately, the northern part of GOODS-S also
K ≃ 25.5)isappropriatetomaintaintheerrorinthemassesti- includesaclusteratz≃ 1.6(Castellanoetal.2007)andvarious
S
mate(seenextsection)towithin0.3dexandtherelativescatter groupsatz≃2.2−2.3(Salimbenietal.2009a;Yangetal.2010;
in the photometric redshifts |∆z|/(1+z) < 0.1 for 85% of ob- Magliocchettietal.2011),whichbothaffecttheoverallredshift
jects, and we adopt this in the following. From the analysis of distribution.
theindividualphotometric-redshiftprobabilitydistributions,we AnotherdifferencefromtheFontanaetal.(2006)analysisis
cancomputethefractionof”reliable”candidates.Weconsidered that the final sample used in this work includesType 2 AGNs,
acandidatetosafelyliewithinagivenredshiftintervalwhenthe sinceweshowinSantinietal.(2012)thattheirstellarmasses-
integralofitsprobabilitydistributioncurve,normalizedtounity, timate is insignificantly affected by the nuclear emission. The
over that interval is larger than 90%. Moreover,we accepted a sameisuntrueforType1AGNs(Santinietal.2012),sowere-
certain level of tolerance in the definition of the redshift range movedspectroscopicallyidentifiedType1AGNsfromthesam-
toallowfortheuncertaintyinphotometricredshifts.Following ple. Since their number is very small (only four sources iden-
this method, for all K < 25.5 sources with z > 2, we can tified in the entire sample), their removal does not affect the
S phot
excludeasecondaryredshiftsolutionatz < 1.5in97.2%of GSMF estimate. We also removed all identified Galactic stars.
phot
the sources. This fractionincreases to 99.6%when only bright Finally,weappliedaredshiftselectiontotherange0.6−4.5and
sources(K < 24)areconsidered.Wealsoextracteda K band weendedupwithasampleof2709objects(ofwhich354have
s S
detectioncatalogue,andverifiedthatalltheobjectsdetectedin spectroscopicredshifts).
3
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
robustbecauseofacombinationofeffectsintheestimateofthe
galaxy star formation rates and ages. Moreover, τ-models are
widelyusedeveninthemostrecentliteratureandallowadirect
comparisonwithpreviousworks.
We adopted a Salpeter IMF. We also computed the stellar
masses by assuming a Chabrier IMF, and checked that these
are simply shifted by a factor -0.24 dex, which is constant to
within3%atthedifferentredshifts.Moreover,wetestedthatthe
GSMFsobtainedbyadoptingthetwoIMFsareconsistentafter
applyingthe same shift, in agreementwith what was foundby
Salimbenietal.(2009b).
The comparison with our previous GOODS-MUSIC sam-
ple allows us to test the effect of a deeper photometry data set
on the accuracy of photometric redshifts and stellar masses.
For this reason, we compared photometric redshifts and stel-
lar masses for identical objects. The photometric redshifts of
thepresentGOODS-ERSdataareinverygoodagreementwith
the GOODS-MUSIC ones. Considering all objects in common
between the two catalogues, the average scatter is < |z −
ERS
z |/(1+z ) >= 0.07,with only0.06%ofsevere
GOODS−MUSIC ERS
(scatter > 0.5) outliers. The stellar masses are also consistent
with those derivedfromthe GOODS-MUSICcatalogue.When
selectinggalaxiesforwhichthe redshiftestimate differsby 0.1
at most, the scatter (M − M )/M is on av-
ERS GOODS−MUSIC ERS
Fig.2.UpperDistributionofthe∆M/Mratio,where∆Misthe erage equalto −0.03±0.40.The majorimprovementprovided
average1σerrorbar foreach object,at z ∼ 2 (left) and z ∼ 3 by the higher quality photometry of WFC3 observations leads
(right). Black solid histogramsrefer to the GOODS-ERS sam- to a reductionin the uncertaintiesin the stellar masses. In Fig.
ple,whereasreddottedonesrepresenttheGOODS-MUSICdata 2, we compare the relative error in the stellar mass estimated
set. Lower Median ∆M/M in each of the redshift bins used in usingtheGOODS-ERSdataset(blacksolidcircles/solidlines)
thisworkasafunctionofthecentralredshiftforGOODS-ERS with that obtained from GOODS-MUSIC catalogue (red open
(blacksolid circles/solidlines) andGOODS-MUSIC(redopen boxes/dottedlines).Weagainselectedonlygalaxiescommonto
boxes/dottedlines). bothcatalogueswithconsistentredshifts.Intheupperpanels,we
showthedistributionofthe∆M/Mratio,where∆Mistheaver-
age1σerrorbarforeachobject(∆M =(M −M )/2),in
∗max ∗min
3. Thegalaxystellarmassfunction(GSMF) two redshiftintervals centred at z = 2 and z = 3; in the lower
panel, we have plotted, as a function of redshift, the median
3.1.Stellarmasses
∆M/Mineachoftheredshiftbinsusedinthiswork.Therelative
Stellar masses were estimated by fitting the 14 band photom- errorsinthestellarmassfortheGOODS-ERSsampleareonav-
etry (up to the 5.5 µm rest-frame) to the Bruzual & Charlot erage∼ 30%smallerthanthosecomputedforthesameobjects
syntheticmodels,inboththeir2003(BC03hereafter)and2007 using the GOODS-MUSIC photometry, the difference increas-
version(Bruzual2007, CB07), througha χ2 minimization.For ingwithredshift.Itisclearthatdeepphotometryinthenear-IR
consistency with our previous works and most of the studies regimeiscrucialtohelpimproveourstellarmassestimates.
in the literature, we adopted the estimates derived with BC03
templates as the reference ones. In the fitting procedure, red-
3.2.TheGSMFestimate
shiftswere fixedtothe spectroscopicor photometricones.Our
1σerrors,causedbyboththephotometricuncertaintiesandthe We estimated the GSMF by adopting both the non-parametric
photometric-redshiftscatter, were computed by consideringall 1/V method (Schmidt 1968) and the STY (Sandageetal.
max
thesolutionswithinχ2 +1.Duringtheerrorcomputation,spec- 1979)maximum-likelihoodanalysisassumingaSchechterpara-
min
troscopicredshiftswere fixedto their value,while photometric metric form. As for any other magnitude-limited sample, our
oneswereallowedtovaryaroundtheirbest-fitsolutioninorder sample does not have a defined limit in stellar mass. For this
toaccountfortheirdegeneracy. reason,ateachstellarmassandeachredshift,wecomputedthe
We parametrized the star formation histories as exponen- fractionofobjectslostbecauseofthelimitedwidthoftheM /L
∗
tiallydeclininglawswithatimescaleτ.Weanalysedawidepa- distributionbyadoptingthetechniquedescribedinFontanaetal.
rameter space for metallicities, ages, extinctions, and τ, whose (2004), after verifying that the simple parametrization used to
details can be found in Fontanaetal. (2004), as updated in describetheobservedM /Ldistributionstillholdsforoursam-
∗
Santinietal. (2009). With respect to our previous works, we ple.
also excluded templates with super-solar metallicity at z ≥ 1. We show the results of our analysis as black solid circles
Studies of the mass-metallicity relation (Maiolinoetal. 2008) (1/V method) and black solid lines (STY approach) in Fig.
max
indeeddemonstratedthatgalaxiesat highredshiftare typically 3 (where the reference BC03 templates were used). The best-
characterizedbysub-solarmetallicities.Wedecidedtoadoptex- fit Schechter parameters are reported in Tab. 1. The error bars
ponentiallydeclining τ models despite it being likely that they in the 1/V points include Poissonian uncertainties, as well
max
areapoorandoversimplifieddescriptionforthestar formation asuncertaintiesinthe stellar masses. Thelatter wereestimated
history (e.g., Marastonetal. 2010). However, Leeetal. (2010) bymeansofaMonteCarlosimulation,wherewerandomlyex-
showed that the resulting stellar masses can still be considered tractedthestellarmassesaccordingtotheir1σuncertaintiesand
4
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
Fig.3.GSMFsobtainedwithBC03stellartemplatesindifferentredshiftrangescomparedwithpreviousworks.Blacksolidcircles
represent our analysis with the 1/V method, black solid lines show the best-fit to a Schechter function according to the STY
max
approach.Thegreydottedlinereplicatesthebest-fitSchechterfunctionatz ∼ 0.8inthehigherredshiftpanels.Errorbarsinclude
the uncertaintiesin thestellar masses aswellasPoissonianerrors.Thehighestmasspointsare oftenpoorlydetermined,because
of the large statistical error, resulting from our poor sampling of the massive side. Other symbols represent the 1/V results
max
of previousworks,scaled to the same cosmologyand convertedto the same IMF: Fontanaetal. (2006):open bluecircles (F06);
Bolzonellaetal. (2010): solid green triangles (B10); Pozzettietal. (2010): open purple triangles (P10); Ilbertetal. (2010): grey
stars(I10);Kajisawaetal.(2009):orangecrosses(K09);Marchesinietal.(2009):solidredboxes(M09);Marchesinietal.(2010):
openpinkboxes(M10);Mortlocketal.(2011):soliddarkgreenpentagons(Mo11);Gonza´lezetal.(2011):openvioletpentagons
(G11).Allthe literatureworksconsideredforthecomparisonadoptedthesame stellar templatesasthisstudy.The legendshows
theredshiftintervalsinwhicheachsetofpointswascomputed.
re-computed the GSMF of 10000 mock catalogues using the secondary photometric-redshiftsolution falls outside each red-
sameproceduredescribedabove. shift interval. This fraction was defined following the criterion
Giventhedegeneracybetweenthefaint-endslopeαandthe discussed above and allowing a tolerance in photometric red-
characteristic mass M∗ of the Schechter function, the STY ap- shift of 0.2. We found that the number density, given by the
proach suffers from the incomplete sampling of the high mass normalization parameter φ∗, decreases with increasing redshift
regimeowingtooursmallarea,especiallyathighredshift.For from10−3.70+−00..0067 atz ∼ 0.8to10−4.12+−00..0180 atz ∼ 4.We recallthat
thisreason,thehighestmass1/V pointsareoftenpoorlyde- cosmic variance effects could cause oscillations in the normal-
max
termined,becauseofthelargestatisticalerror.Inthehighestred- ization parameter, especially in the two bins that are most af-
shiftbin,toconstrainthefit,wefixed1 M∗ tothevalueobtained fectedbythepresenceofoverdensities,namelythe1.4−1.8and
atz ∼ 3.ThefitsfoundinthiscaseareshowninFig.3andthe 1.8−2.5redshiftintervals.However,asimilarlydecreasingtrend
relevantSchechterparametersaregivenin Tab.1.Inthistable, forthenormalizationwasalsoobservedbypreviousworks(e.g.
we also provide (third column) the fraction of objects where a Fontanaetal.2006; Pe´rez-Gonza´lezetal.2008;Kajisawaetal.
2009;Marchesinietal.2009;Mortlocketal.2011).Mostinter-
1 InSect.4,westudyhowthebest-fitparameterαvarieswhenchoos- estingly,thelow-massslopesteepenssignificantlyfromz ∼ 0.8
ingadifferentvalueofM∗. to z ∼ 3, where the Schechter parameter α decreases from
5
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
−1.44±0.03to−1.86±0.16,andthenflattensfromz∼3toz∼4. quality. They also did not include any K band data, which is
AsdemonstratedinSect.4,thisresultremainsvalidheredespite importantto estimate reliable stellar masses. Finally, since our
the uncertaintiesderivedforthe smallarea coveredby oursur- study is based on 14 bands of photometry (instead of 6 bands
vey and the presence of known overdensities. Indeed,even the as Mortlocketal. 2011), our work also relies on good quality
redshift ranges that are the most contaminated show faint-end photometricredshifts.
slopesinlinewiththeresultsintheotherredshiftbins. Despitethelimitedskyarea,thebright-endtailiscompara-
Up to z ∼ 2.5, the GSMF shows a dip at M⋆ ≃ 1010M⊙, ble overall within the uncertainties with that inferred by large
which seems to shift to higher stellar masses as redshift in- surveysoverthewholeredshiftrange(withtheexceptionofthe
creases,implyingthatasingleSchechterisapoorparametriza- 1.4−2.5redshiftinterval,which,asdiscussedabove,isaffected
tion. This dip has been identified in previous wide-field sur- bythepresenceofoverdensities).Theonlyseveredisagreement
veys and interpreted as the differential evolution of the red isfoundwhencomparingourresultsinthehighestredshiftinter-
andthebluepopulations(Ilbertetal.2010;Pozzettietal.2010; valtoGonza´lezetal.(2011),who,asalreadypointedoutinthe
Bolzonellaetal.2010;Mortlocketal.2011).Theeffectislarger introduction,derivedtheGSMFbyusingadifferentprocedure,
in the redshift intervals 1.4 < z < 1.8 and 1.8 < z < 2.5, i.e. by combining the UV luminosity function with an average
whicharehighlyaffectedbythepresenceofawell-knownclus- M /Lratio.
∗
ter at z ∼ 1.6 (e.g. Castellanoetal. 2007) and of several of lo-
calized overdensities at z ≃ 2.2−2.3 (Salimbenietal. 2009a;
Yangetal. 2010), respectively:they are indeed populated by a 3.4.Theeffectofdifferentstellartemplates
higher fractionof old red galaxies, which enhancesthis dip. A
Thesystematicuncertaintiescausedbythevariousassumptions
differentexplanationofthediparound∼1010M wassuggested
⊙ involved in spectral energy distribution modelling were shown
by Droryetal. (2009), who also measured a bimodal shape in
to dominate the overall error budget affecting the GSMF (see
the GSMF of the blue and red populationsseparately. This di-
Marchesinietal.2009,foradetailedanalysis).Inthisregard,a
chotomyingalaxyformation,whichpre-datestheredsequence
significant role is played by the choice of the stellar templates
appearance,wasascribedtoachangewithstellarmassineither
usedtoestimatethestellarmass.
star formation efficiency or galaxy assembly rate. The studies
StellarmassesobtainedusingtheCB07stellarlibrary,which
citedaboveshowthatadoubleSchechterisamoreaccuratede-
includes an improved TP-AGB stars treatment, are on average
scription of the shape of the total GSMF. However, given the
0.12 dex lower than those inferred using the BC03 templates,
smallsizeofoursample,theinclusionoftwomorefreeparam-
with a scatter as largeas0.17dex.We plotin Fig. 4 theirratio
eters (Bolzonellaetal. 2010) makes the fit degenerateand was
as a function of the stellar mass adopted as a reference in this
notanapproachthatweadoptedhere.
work(M )indifferentredshiftbins.Thelackofacleartrend
BC03
of M /M with stellar mass or redshift translates into a
BC03 CB07
3.3.Comparisonwithpreviousresults lack of a rigid offset between the GSMFs computed with the
twolibraries,althoughtheCB07pointsareonaverageatlower
WeshowinFig.3acompilationof1/V pointscollectedfrom
max stellarmassesthanBC03.
the literature, as listed in the legend, scaled to the same cos-
We compare in Fig. 5 the GSMFs obtained with the BC03
mology and IMF. Unfortunately,it is impossible to correct for
templates (black solid curves/solid circles) and the CB07 ones
the effects of different stellar libraries, because, as we show in
(red dotted curves/open boxes). For the sake of simplicity, we
the next section, we are unable to determine a systematic shift
decidedto reportthe four mostrepresentativebins. The results
that they could cause in the GSMF. Therefore, we decided to
for the 1.0−1.4and 1.4−1.8redshiftbinsare verysimilar to
showonlythosestudiesthatadoptthesamestellarlibraryasthis
the 0.6 − 1.0 and 1.8 − 2.5 ones, respectively. We also show
work. Overall, our results are in good agreement with most of
the 1/V points of Marchesinietal. (2009) (their set 8) and
the other surveys, especially up to z ∼ 3. In the two redshift max
Caputietal. (2011), both obtained by adopting the CB07 tem-
intervals affected by the overdensities (z ∼ 1.6 and z ∼ 2.2),
plates. The results of Marchesinietal. (2009) agree with our
our GSMFs are on average higher than the other surveys, but
CB07-based GSMF in all except the 1.8 − 2.5 redshift inter-
still consistentwith most of them within the errors.In general,
val,likelybecauseofimperfectredshiftoverlapbetweenthetwo
we report a larger number of galaxies at the bright tail than
analysis.ThepointsfromCaputietal.(2011)areinbroadagree-
Fontanaetal.(2006),becausethepresentstudyincludesAGNs
mentwithoursatthebrightend,whiletheincompletenessthat
(except the few identified Type 1), which preferentially live
theauthorsclaimtobeaffectedbybelowM ∼1011M islikely
inhighmassgalaxies(Bundyetal.2008;Alonso-Herreroetal. ∗ ⊙
responsibleforthedisagreementatlowstellarmasses.
2008;Brusaetal.2009;Silvermanetal.2009;Xueetal.2010;
Santinietal. 2012). Given the very deep near-IR observations Thebest-fitSchechterparametersoftheCB07-basedGSMF
used in this work, the sampling of the low-mass end of the arereportedinTab.2.Atz>2.5,wewereforcedtofix2theM∗
GSMFisconsiderablyfinerthanmostprevioussurveys,onav- parametertoitsbest-fitvalueatz∼2.15.Ifitisinsteadallowed
erageby0.5dexuptoz∼1.8andby0.1dexatz∼4,andatthe tovary,thefitisunconstrainedorthemaximum-likelihoodanal-
sametimetheconservativephotometriccut(K <25.5)ensures ysisdoesnotconverge.TheCB07-andBC03-basedGSMFsdif-
S
reliableresultsevenatthelowestmasses. fer from each other. However, we do not find any similar sys-
The only comparable study sampling similar or slightly tematicbehaviouratallredshifts.Thatthehigh-massendofthe
lower stellar masses is the one of Mortlocketal. (2011). This CB07-basedGSMFisunconstrainedatz>2.5,whiletheBC03-
work is somewhat peculiar, being obtained from a set of bi- basedonesuffersfrompoorstatisticalsamplingonlyinthehigh-
ased pointings specifically designed to contain as many mas- estredshiftbin(z > 3.5),is a furtherconfirmationthatthe two
sive galaxies as possible, and a posteriori corrected to account GSMFsarenotaffectedbyasystematicshiftinstellarmass.
for this bias. They pushed their detection to H = 26.8 at a 5σ
level,whileoursample,althoughextractedfromimagesofsim- 2 SeeSect. 4for an analysis of how thebest-fit parameter α varies
ilardepth,wascutatabrighterlimittoensuregoodphotometric whenchoosingadifferentvalueforM∗.
6
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
Fig.5. Comparison between the GSMFs obtained with the
Bruzual&Charlot (2003) template library (black solid circles
and solid curves) and the Bruzual (2007) one (red open boxes
anddashedcurves).Symbolsaretheresultsofthe1/V anal-
max
ysisandcurvesrepresenttheSTYSchechterfits.Errorbarsin-
cludetheuncertaintiesinthestellarmassesaswellasPoissonian
errors.Othersymbolspresent1/V resultsofpreviousworks
max
based on CB07 templates, scaled to the same cosmology and
Fig.4.RatioofstellarmassescomputedwithBC03(MBC03)and convertedtothesameIMF:Marchesinietal.(2009):solidpur-
CB07(MCB07)templatesversusMBC03indifferentredshiftbins. pletriangles(M09);Caputietal.(2011):cyanstars.
4. Thefaint-endslope
At the lowest and the highest redshifts, we find the closer
agreement,the normalizationof the best-fit Schechter function Themaingoalofthisstudyhasbeentoinvestigatethefaint-end
beingonlyslightlylowerwhenCB07templatesareused.Atin- slope of the GSMF, especially at the highest redshifts (z > 2).
termediate redshifts, we observe a more serious disagreement, FrombothFig.3andTables1and2,itisevidentthatthelow-
resultingindifferentfaint-endslopesandcharacteristicmasses. mass tail steepens with redshift. The results from applying the
Thisis unsurprising,because the effect of includingof the TP- STYapproachtoourBC03-baseddataindicatethatthefaint-end
AGB phase is expected to be important at intermediate ages slope steepens significantly between z ∼ 0.8, where we fitted
(0.2−2 Gyr),whichpredominatethe 2 . z . 3redshiftrange α = −1.44±0.03, and z ∼ 3, where the best-fit α is equal to
(Maraston2005;Henriquesetal.2011).Althoughthedifference −1.86±0.16,beforeflatteninguptoz∼4.
betweentheCB07-andtheBC03-basedGSMFsdonotshowa Firstofall,weperformedasimplesanitychecktoverifythat
systematic trendatallredshifts,the characteristicmasses seem theabundanceoflowmassobjectsatz>1.8isreliablebyplot-
tobeonaveragelowerwhenCB07templatesareused(seeFig. tingallobjectswith M < 1010M and1.8 < z < 2.5ona BzK
∗ ⊙
6 discussed in the next section), as expected, despite the large diagram.Forgalaxiesatz>2.5,weadoptedtheanalogousRJL
uncertainties,in agreementwith the resultsof Marchesinietal. diagram(usingIRAC3.6µmasLband),whichextendsthefor-
(2009) . This trend is clear at 1.4 < z < 2.5, where our red- mertothe2.5 < z < 4redshiftregime(Daddietal. 2004),and
shift bins overlap with those of Marchesinietal. (2009), while checked stellar masses below 2·1010M . Approximately91%
⊙
thelackofhighqualitystatisticsathigherredshiftspreventsus of the sources indeed lie in the high redshift regions of these
from drawing any firm conclusions about the effect of chang- diagrams,makingusconfidentoftheirphotometricredshiftesti-
ing the stellar templates. For what concerns the variation in α mate.Asanadditionalcheck,wecarefullyinspectedtheindivid-
whenchangingthetemplatelibrary,wefindsimilarslopesfrom ualphotometric-redshiftprobabilitydistributioncurvesforeach
z ∼ 0.8 to z ∼ 1.2, while in the redshift interval 1.4−2.5 the sourcewithz>1.8andM <2·1010M .Followingthecriterion
∗ ⊙
BC03-based GSMFs are steeper than the CB07-based ones by describedinSect.2,wefoundthat96.5%ofthesesourceshave
0.2−0.3. Marchesinietal. (2009) reports similar slopes when a90%probabilityoflyingatz>1.5.
using BC03 and CB07 stellar templates in the redshift interval AsalreadypointedoutinSect.3.2,thesmallskyareasam-
1.3 − 3.0, while their BC03-based GSMF is steeper than the pled by our data may be responsible for degeneraciesbetween
CB07-based one at 3.0 < z < 4.0. The intrinsic difference be- the faint-end slope α and the characteristic mass M∗ when fit-
tweenthetwosurveysdoesnotallowustoinvestigatetheorigin tingaSchechterfunction.Wethereforestudiedindetailthede-
ofthismismatch. generaciesintheα−M∗ plane.TheresultsareshowninFigs.6
7
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
Fig.6. The α and M∗ space (1σ and 2σ contours) resulting Fig.7. Best-fit M∗ andαmodelparametersobtainedbyadopt-
fromthemaximum-likelihoodanalysis.Blacksolidcurvesrefer ing BC03 (upper panel) and CB07 (lower panel) templates at
to BC03-basedGSMFs, red dotted curvesreferto CB07-based differentredshifts.Thesymbolsizeincreaseswithredshift.The
ones. shadedareasshowthe valuesofthe faint-endslope atdifferent
fixedM∗intheredshiftbinsindicatedbythelabels.
and7.Inthefirstfigure,weanalysedtheredshiftintervalswhere
both parameterswere allowed to vary,while in the secondone
westudiedthedependenceofthebest-fitαonthechosenM∗ in faint-endslopeatdifferentfixedM∗intheredshiftbinsindicated
thoseredshiftbinswherewewereforcedtofixthecharacteristic bythelabels.Thesymbolsshowthebest-fitvalues(andtheirun-
masstoconstrainthemaximum-likelihoodanalysis. certainties)forαand M∗ atz < 3.5(z < 2.5whenusingCB07
In Fig. 6, we show the 1σ and 2σ contours for α and templates),whereourresults are insignificantlyaffectedby the
M∗ Schechter parameters, for both the BC03-based GSMFs lackofhighqualitystatistics andwecouldallowbothparame-
(black solid curves) and CB07-based ones (red dotted curves). terstovary.TheupperpanelreferstotheBC03-basedGSMFs,
While the parameter α is well-constrained at all redshifts (al- while the lower one is obtainedby adoptingCB07 stellar tem-
though with uncertainties increasing with z), our data prevent plates.
us from properlyinferringthe value of the characteristic mass. FromFig.7(lowerpanel,blueshadedregion),itappearsthat,
Nonetheless,asweshowbelow,theresultonαisrobustagainst whatever reasonable value for M∗ is chosen at 2.5 < z < 3.5,
thedegeneracyofM∗. the best-fit α is clearly steeper than the best-fit values at lower
The steepening in α between z ∼ 0.8 and z ∼ 3 is clear redshifts,confirmingtheresultfoundwithBC03templatesand
from Fig. 6 when the BC03 stellar library is used. When we supporting that the major result of this paper is unaffected by
instead adopted CB07 templates, the faint-end slope did not thepoorsamplingofthehighmassregime.However,asshown
changemuchfromz ∼ 0.8toz ∼ 2.2,whileathigherredshifts bythe red shadedregionsin Fig. 7, presentlyavailable datado
(2.5<z<3.5)wewereforcedtofixthevalueofM∗toconstrain not allow us to draw any firm conclusion about the trend of α
thefit(seeSect.3.4),makingthebest-fitαparameterdependent betweenz∼3andz∼4.
onthechoiceofthecharacteristicmass. Tocorroborateourresultfortheslopeofthelow-massend,
To study how the best-fit value of α changeswhen varying andensurethatisunaffectedbythelackofstatisticsatthemas-
theM∗ value,webuiltagridoflog(M∗[M ])rangingfrom10.5 sive end, we took advantage of the outcomes of large surveys
⊙
to11.6(theselimitsarejustifiedbypreviousworks)withsteps and followed different approaches.We fitted the 1/V points
max
inmassof0.1andwefittedaSchechterfunctiontothedatafor fromthisstudytogetherwiththosecollectedfromtheliterature
each element of the grid. We adopted this procedure in all the in comparableredshiftintervals. We note thatin principle, and
redshiftsbinswherethefitisunconstrained.WeshowinFig.7 in contrastto the STY approach,fitting 1/V pointsinvolves
max
theα−M∗plane,wheretheshadedareasshowthevaluesofthe databinning,thusmayingeneralproduceadifferentfit.Wein-
8
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
Thetendencyforthelow-massendoftheGSMFtosteepen
withredshiftwaspreviouslyfoundbyotherauthors.According
to the evolutionary STY fit of Fontanaetal. (2006), α ranges
from−1.25±0.03atz∼0.8to−1.51±0.13atz∼4.Increasing
with redshiftbut flatter values for the faint-end slope were ob-
tained by Marchesinietal. (2009), who reported α = −0.99 at
z ∼ 1.6 and α = −1.39 at z ∼ 3.5. Kajisawaetal. (2009)
found α = −1.26+0.03 at z ∼ 0.75 and α = −1.75+0.15 at
−0.03 −0.13
z ∼ 3. Very steep GSMFs, in agreement with those inferred
withourdata,werefittedbyCaputietal.(2011),whomeasured
α = −2.07+0.08 at z ∼ 3.9. Finally, Mortlocketal. (2011) re-
−0.07
portedα = −1.36±0.05 at z ∼ 1.25and α = −1.89±0.11at
z ∼ 2.75. Both Mortlocketal. (2011) and Caputietal. (2011)
found a flattening at higher redshift similar to the one that we
measureatz ∼ 4,i.e. theyfitted α = −1.73±0.09at z ∼ 3.25
andα=−1.85+0.27atz∼4.6,respectively.
−0.32
Wenotethatequallyrobustresultscannotbeinferredforthe
evolutionofthecharacteristicmassM∗,whosebest-fitvaluesare
Fig.8. Faint-endslope as a functionof redshift.The parameter highlysensitivetothestellartemplatesandthefunctionalshape
α was computed through a maximum-likelihood analysis with fittedtothedata(seeTabs.1,2and3),aswellasthesizeofthe
a Schechterform(black solid circles referto the BC03 library, sample.
red open boxes refer to the CB07 one) and by fitting the en-
semble (this study + previous surveys) of 1/V points with
max
5. Thestellarmassdensity(SMD)
aSchechterparametricform(blueopentriangles)andadouble
power-lawshape(seetext,greenopencircles).Thedifferentsets We computed the total SMD by integrating the analytical fit-
areshifted inredshiftwith respectto thecentralvaluesineach ting functions in each redshift bin from 108 to 1013M . We
⊙
interval(shownbythesolidblackcircles). show in Fig. 9 the SMD derived from the STY analysis using
BC03andCB07templatesassolidblackandredcurves,respec-
tively. We also show a compilation of results from the litera-
cluded only those surveys whose results are obtained using a ture, reported to the same cosmology and IMF, as listed in the
methodsimilartoourownandthatsamplethehigh-masstailof legend. The same integration limits as in this study were used
thedistribution,typicallyaboveM∗ ≃3×1010M⊙.However,we in most of the works considered. The only exceptions are the
obtainedverysimilarresultswhenalsoincludingthepointsfrom Mortlocketal. (2011) points (M > 107M ), the Ilbertetal.
∗ ⊙
theliteratureatlowermasses.WefoundthatasingleSchechter (2010) ones (M > 105M ), and those from Dickinsonetal.
∗ ⊙
function does not seem to reproduce the faint- and the bright- (2003)andPe´rez-Gonza´lezetal.(2008),whoadoptedredshift-
end simultaneously in a satisfactory way. This is unsurprising dependentmasslimits(werefertotheseworksformoredetails).
becausetheSchechterfunctionisitselfapoordescriptionofthe Ourresultsshowgoodagreementwith thosecomputedbypre-
shape of the GSMF when samples with high quality statistics vious authors at 0.6 . z . 2, although we recall once again
are used (see discussion in the introduction and in Sect. 3.2). that our mass densities in the redshiftintervals aroundz ∼ 1.6
However,theinhomogeneityofthedatasetcanalsoplayarole: and z ∼ 2.2−2.3 mightbe systematically too high owing to a
we collected 1/V pointsfrom differentsurveys, observedin few known overdensities.The steepness in the faint-end of the
max
different sky areas, and computed with slightly different meth- GSMF computed by this work is responsible for the large val-
ods. We then fitted the ensemble of the 1/V points from uesoftheSMDinferredatz > 2.Ourestimatesarehigherthan
max
thisworkplusthosecollectedfromtheliteraturewith adouble those reported by previous authors, with the exception of the
power-law3. The best-fit parameters are shown in Tab. 3. This Mortlocketal. (2011) results. However,the latter results origi-
analyticshape,havingoneadditionaldegreeof freedomthana natefromadifferentshapeoftheGSMF:Mortlocketal.(2011)
singleSchechterfunction,providesatighterfittothedataatall indeed found flatter faint-end slopes than we do, and the large
redshifts. SMDisaconsequenceofahigherdensityofhighmassgalaxies
Wereportthedifferentvaluesofthefaint-endslopeasafunc- (seeFig.3).
tionofredshiftinFig.8.Itisshownthat,regardlessofthestellar WenotethattheSMDisaffectedbyuncertaintiescausedby
templatesandmethodadoptedandthefunctionalshapefittedto systematic effects. In Fig. 9, the grey shaded region indicates
thedata,alltheresultsindicateasteepeningofthefaint-endof thedispersionintheSMDwhenincludingtheoutputsobtained
theGSMFwithredshiftuptoz ∼ 3.Thetrendisrobustdespite by integrating the fit to our 1/V points plus those collected
max
therelativelylargeerrorbars,especiallyathighredshift,andthe fromtheliteraturewithbothaSchechterfunctionandadouble
presence of known overdensitiesat z ∼ 1.6 and z ∼ 2.2−2.3, power-law shape (see Sect. 4). This region represents the sys-
and it is unaffected by the lack of high quality statistics at the tematicerrorscausedbythechoiceofthestellarlibraryandthe
high-massendtypicalofsmallskyareas.Thesteepeningofthe functionalshape ofthe GSMF, as wellas the simultaneoususe
faint-endslopewithredshiftseemstohaltatz>3andthevalue of the ERS observationsas a probeof the low-massend of the
of α seems to remainconstantup to z ∼ 4. However,although GSMF and the results of large surveys to constrain the bright-
thisisconfirmedbytheuseoftheoutcomeofpreviouslargesur- end. The dispersion increases significantly at z & 3, reflecting
veys, the results based on our data alone are largelydependent the largescatter amongexistingsurveys.Moreover,the lack of
onthechoiceofthefixedM∗ parameter. overlap with most previous results at these redshifts is a sign
oftheimpossibilitytoassembleasingle,self-consistentGSMF
3 Theassumedfunctionalshapeisφ /[(M/M∗)−(1+α)+(M/M∗)−(1+β)] fromthehighesttothelowestmasses.Thisisduetotheinhomo-
∗
9
P.Santinietal.:Theevolvingslopeofthestellarmassfunctionat0.6≤z<4.5fromdeepWFC3observations
Schechterparameters(STYmethod)–BC03templates
Redshiftbin N % α log M∗(M ) log φ (Mpc−3) logρ(M Mpc−3)
notsecure 10 ⊙ 10 ∗ ⊙
0.6-1.0 584 8.5% -1.44±0.03 11.67±0.17 -3.70+0.06 8.51
−0.07
1.0-1.4 375 6.2% -1.47±0.05 11.61±0.18 -3.47+0.07 8.35
−0.08
1.4-1.8 259 9.7% -1.60±0.07 11.75±0.26 -3.85+0.11 8.23
−0.14
1.8-2.5 425 7.4% -1.84±0.06 11.82±0.28 -4.17+0.14 8.29
−0.20
2.5-3.5 182 11.8% -1.86±0.16 11.30±0.27 -3.94+0.16 7.97
−0.26
3.5-4.5 51 8.4% -1.80±0.20 11.30(fixed) -4.12+0.08 7.72
−0.10
Table 1.Best-fit Schechterparametersin thedifferentredshiftintervalsasa resultofthe STYapproachusingBruzual&Charlot
(2003)templates.Parameterswithnoerrorbarshavebeenfixedtothevalueinthelowerredshiftbin.Thesecondcolumnindicates
thenumbersofgalaxiesineachredshiftbinbasedonwhichtheGSMFisactuallycomputed.Thethirdcolumnshowsthefraction
ofgalaxieswhereasecondaryredshiftsolutionoutsidetheredshiftbincannotbediscardedwitha90%probability(seeSect.3.2
fordetails).ThelastcolumnreportsthecorrespondingmassdensityρobtainedbyintegratingtheGSMFbetween108and1013M .
⊙
Schechterparameters(STYmethod)–CB07templates
Redshiftbin N % α log M∗(M ) log φ (Mpc−3) logρ(M Mpc−3)
notsecure 10 ⊙ 10 ∗ ⊙
0.6-1.0 509 8.5% -1.50±0.03 11.85±0.24 -3.70+0.10 8.39
−0.13
1.0-1.4 372 6.2% -1.46±0.05 11.50±0.18 -3.49+0.08 8.22
−0.09
1.4-1.8 264 9.7% -1.42±0.07 11.32±0.17 -3.41+0.09 8.09
−0.12
1.8-2.5 437 7.4% -1.58±0.06 11.29±0.15 -3.52+0.10 8.07
−0.12
2.5-3.5 153 11.8% -2.16±0.11 11.29(fixed) -4.39+0.28 8.05
−1.14
3.5-4.5 45 8.4% -1.88±0.21 11.29(fixed) -4.28+0.11 7.67
−0.15
Table2.SameasTab.1usingBruzual(2007)templates.
Doublepower-lawparameters(fitto1/V points)–BC03templates
max
Redshiftbin α β log M∗(M ) log φ (Mpc−3) logρ(M Mpc−3)
10 ⊙ 10 ∗ ⊙
0.6-1.0 -1.36±0.02 -4.47±0.12 11.39±0.01 -2.75+0.02 8.50
−0.02
1.0-1.4 -1.52±0.02 -5.24±0.16 11.38±0.01 -3.10+0.02 8.24
−0.02
1.4-1.8 -1.49±0.05 -4.70±0.23 11.30±0.03 -3.12+0.05 8.12
−0.06
1.8-2.5 -2.01±0.04 -6.25±1.57 11.64±0.06 -3.94+0.09 8.28
−0.12
2.5-3.5 -2.28±0.08 -6.70±4.84 11.77±0.10 -4.73+0.17 8.24
−0.28
3.5-4.5 -2.27±0.25 -6.38±7.48 11.81±0.19 -4.84+0.34 8.15
−4.84
Table3.Best-fitparametersofthedoublepower-lawshapefitto1/V pointsfromthiswork(usingBC03templates)+acollection
max
fromtheliteratureatM &3×1010M (seetext).Thelastcolumnreportsthecorrespondingmassdensityρobtainedbyintegrating
∗ ⊙
theGSMFbetween108and1013M .
⊙
geneity of the samples, to the variance between differentfields the z ∼ 2.1 redshift interval), especially when considering the
andalsototheintrinsicuncertaintiesathighredshiftinboththe dispersion caused by the inclusion of high mass 1/V points
max
stellarmassesandGSMF. fromthe othersurveys.Consistencyathigh redshiftwas found
by Mortlocketal. (2011) and Papovichetal. (2011), the latter
We compared the SMD with the integrated star forma-
studybeingbasedonanindependentsanalysis.Overall,ourre-
tion rate density. For this purpose, we first considered the
sultssupportthenotionthattheSMDcanbereasonablycloseto
best-fit to the compilation of SFRD measurements made by
theintegratedSFRDatz > 2,mostlyduetoasteepeningofthe
Hopkins&Beacom(2006).FollowingWilkinsetal.(2008),we
GSMF,althoughourresultsmightbesystematicallytoohighbe-
rescaled it to a Salpeter IMF and integrated it as a function
causeoftheknownoverdensitiesinthesmallERSfield.Asmen-
of time, after accounting for the gas recycle fraction. The lat-
tioned, the higher values that we obtained than most previous
ter is the fraction of stellar mass returned to the interstellar
studies is essentially due to the efficiencyof WFC3 deep near-
medium as a function of time, and was computed using the
IR data to accuratelyrecoverthe faint-endof the GSMF, espe-
Bruzual&Charlot (2003) model. The result of this calculation
ciallyathighredshift,whichcontributessignificantlytothetotal
is shown in Fig. 9 by the blue dashed line. We then performed
SMD.However,thesignificantsteepeninginthefaint-endslope
the same calculation by using the best-fit parametric shape for
presented in this work is insufficientto solve the disagreement
the star formation history inferred by Reddy&Steidel (2009),
atz<2,wheretheintegratedSFRDexceedstheobservedSMD
whichalso includesmorerecenthighredshiftpointsaswellas
by a factor of ∼ 2−3, even when both of them are integrated
a luminosity-dependent dust correction to the z > 2 data. We
down to low values of stellar mass / luminosity and adopting
obtainedthegreenlongdashedlineshowninFig.9.
the SFRD computed by Reddy&Steidel (2009). Their SFRD,
OurresultssolvethediscrepancybetweentheSMDandthe
althoughitislowerthanthatresultingfromthebest-fitrelation
integrated SFRD at z > 2 (modulo the uncertainties affecting
10